Maximal density, kinetics of deposition and percolation threshold of loose packed lattices

Abstract

Using Monte Carlo (MC) simulation governed by dynamic rules we study the kinetics of filling of square lattice under condition that occupied sites prefer to not share edges. This we quantify by introducing certain probability 1≥p≥0 and study its influence on the kinetics of the process as well as on the properties of the obtained systems. In the particular case p=0 the occupied sites cannot share edges (nearest neighbors occupations are not permitted) and we find that the maximal achievable concentration when the sites are chosen at random is Cmax=0.3638±0.0003, well below 0.5 – the concentration of the perfect checkerboard. On the other hand, for any p>0 the occupied sites can share edges, although with hindrance, and Cmax can be exceeded. This is realized by the following MC procedure: an unoccupied site is chosen at random and if it has no neighbors, i.e. Nb=0, it is occupied. If Nb>0 this could happen with probability p. We elucidate how the site percolation threshold Pc value depends on the probability p. With this approach we address examples that may be found in statistical physics, chemistry, materials science, discrete mathematics, etc. For instance, this is the case when particles are attracted to an interface but repulse each other.